Let I(t)=∮δ(t)​ω be an Abelian integral, where
H=y2−xn+1+P(x) is a hyperelliptic polynomial of Morse type, δ(t) a
horizontal family of cycles in the curves {H=t}, and ω a polynomial
1-form in the variables x and y. We provide an upper bound on the
multiplicity of I(t), away from the critical values of H. Namely: $ord\
I(t) \leq n-1+\frac{n(n-1)}{2}if\deg \omega <\deg H=n+1.Thereasoninggoesasfollows:weconsidertheanalyticcurveparameterizedbytheintegralsalong\delta(t)ofthen‘‘Petrov′′formsofH(polynomial1−formsthatfreelygeneratethemoduleofrelativecohomologyofH),andinterpretthemultiplicityofI(t)astheorderofcontactof\gamma(t)andalinearhyperplaneof\textbf C^ n.UsingthePicard−Fuchssystemsatisfiedby\gamma(t),weestablishanalgebraicidentityinvolvingthewronskiandeterminantoftheintegralsoftheoriginalform\omegaalongabasisofthehomologyofthegenericfiberofH.Thelatterwronskianisanalyzedthroughthisidentity,whichyieldstheestimateonthemultiplicityofI(t).Still,insomecases,relatedtothegeometryatinfinityofthecurves\{H=t\}
\subseteq \textbf C^2,thewronskianoccurstobezeroidentically.Inthisalternativeweshowhowtoadapttheargumenttoasystemofsmallerrank,andgetanontrivialwronskian.Foraform\omegaofarbitrarydegree,weareledtoestimatingtheorderofcontactbetween\gamma(t)andasuitablealgebraichypersurfacein\textbf C^{n+1}.Weobservethatord I(t)growslikeanaffinefunctionwithrespectto\deg \omega$.Comment: 18 page
Here SF denotes the category whose objects are the pairs (X,P)where P is a metrizable ANR-space and X is a closed subset of P, and the morphisms between two objects (X,P) and (Y,Q) are the homotopy classes of mutations f:U(X,P)→V(Y,Q) (where U(X,P) and V(Y,Q) are the complete open neighborhood systems of X in P and Y in Q respectively). So two objects of SF are isomorphic if and only if they have the same shape in the sense of Fox (or Marde sic). The author constructs a covariant functor T from SF to the category C0 of all compact 0-dimensional spaces and continuous maps. This functor allows him to obtain new shape invariants in the class of metrizable spaces. Using this functor T he also constructs new contravariant functors to the the category of metrizable spaces and continuous maps and to the category of groups and homomorphisms. In order to construct T he uses the space of quasicomponents QX of a metrizable space X . Actually he uses the 0-dimensional compactification β0(QX) of QX. The space β0(QX) can be viewed as the 0-dimensional analogue of the Stone-Cech compactification. As a theorem he proves that two 0-dimensional metrizable spaces are of the same shape if and only if they are homeomorphic. This is a generalization in the metric case of a similar result for paracompacta due to G. Kozlowski and the reviewer [Fund. Math. 83 (1974), no. 2, 151-154] because there are metrizable spaces X such that ind(X)=0 but the covering dimension dim(X)>0